Question

multiple - choice questions each have five possible answers (a, b, c, d, e), one of which is correct. assume that you guess the answers to three such questions.
a. use the multiplication rule to find ( p(cww) ), where ( c ) denotes a correct answer and ( w ) denotes a wrong answer.
( p(cww)=\frac{16}{125} ) (type an exact answer.)
b. beginning with ( cww ), make a complete list of the different possible arrangements of one correct answer and two wrong answers, then find the probability for each entry in the list.
( p(cww) ) - see above
( p(wwc)=square )
( p(wcw)=square )
(type exact answers.)

Explanation:

Step1: Calculate the probability of a correct answer ($P(C)$) and a wrong answer ($P(W)$)

Since there are 5 possible answers and 1 is correct, $P(C)=\frac{1}{5}$. Then $P(W) = 1 - P(C)=1-\frac{1}{5}=\frac{4}{5}$.

Step2: Use the multiplication rule for independent events

For independent events $A$, $B$, and $C$, $P(A\cap B\cap C)=P(A)\times P(B)\times P(C)$.

For $P(WWC)$:

$P(WWC)=P(W)\times P(W)\times P(C)=\frac{4}{5}\times\frac{4}{5}\times\frac{1}{5}=\frac{16}{125}$.

For $P(WCW)$:

$P(WCW)=P(W)\times P(C)\times P(W)=\frac{4}{5}\times\frac{1}{5}\times\frac{4}{5}=\frac{16}{125}$.

Answer:

$P(WWC)=\frac{16}{125}$; $P(WCW)=\frac{16}{125}$